(sec:methods:choosing-a-formulation:mass-conservation-approx)= # Mass conservation approximation First, we have to choose how to approximate the conservation of mass: $\nabla \cdot (\rho \mathbf u) = 0$ (see Equation {math:numref}eq:stokes-2). We provide the following options, which can be selected in the parameter file in the subsection Formulation/Mass conservation (see also {ref}parameters:Formulation/Mass_20conservation): - "incompressible": {math} \nabla \cdot \textbf{u} = 0,  - "isothermal compression": {math} \nabla \cdot \textbf{u} = -\rho \beta \textbf{g} \cdot \textbf{u},  where $\beta = \frac{1}{\rho} \frac{\partial \rho}{\partial p}$ is the compressibility, and is defined in the material model. Despite the name, this approximation can be used either for isothermal compression (where $\beta = \beta_T$) or isentropic compression (where $\beta = \beta_S$). The material model determines which compressibility is used. This is an explicit compressible mass equation where the velocity $\textbf{u}$ on the right-hand side is an extrapolated velocity from the last timesteps. - "hydrostatic compression": {math} \nabla \cdot \textbf{u} = - \left( \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_{T} \rho \textbf{g} + \frac{1}{\rho} \left( \frac{\partial \rho}{\partial T} \right)_{p} \nabla T \right) \cdot \textbf{u} = - \left( \beta_T \rho \textbf{g} - \alpha \nabla T \right) \cdot \textbf{u}  where $\beta_T = \frac{1}{\rho} \left(\frac{\partial \rho}{\partial p} \right)_{T}$ is the isothermal compressibility, $\alpha = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T} \right)_{p}$ is the thermal expansion coefficient, and both are defined in the material model. The approximation made here is that $\nabla p = \rho \textbf{g}$. - "reference density profile": {math} \nabla \cdot \textbf{u} = -\frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial z} \frac{\textbf{g}}{\|\textbf{g}\|} \cdot \textbf{u},  where the reference profiles for the density $\bar{\rho}$ and the density gradient $\frac{\partial \bar{\rho}}{\partial z}$ provided by the adiabatic conditions model ({ref}sec:methods:initial-conditions) are used. Note that the gravity is assumed to point downwards in depth direction. This is the explicit mass equation where the velocity $\textbf{u}$ on the right-hand side is an extrapolated velocity from the last timesteps. - "implicit reference density profile": {math} \nabla \cdot \textbf{u} + \frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial z} \frac{\textbf{g}}{\|\textbf{g}\|} \cdot \textbf{u} = 0,  which uses the same approximation for the density as "reference density profile," but implements this term on the left-hand side instead of the right-hand side of the mass conservation equation. This effectively uses the current velocity $\textbf{u}$ instead of an explicitly extrapolated velocity from the last timesteps. - "ask material model," which uses "isothermal compression" if the material model reports that it is compressible and "incompressible" otherwise. :::{admonition} The stress tensor approximation :class: note If a medium is incompressible, that is, if the mass conservation equation reads $\nabla \cdot \textbf{u} = 0$, then the shear stress in the momentum and temperature equation simplifies from {math} \tau = 2\eta\left(\varepsilon\left(u\right) - \frac{1}{3}\left(\nabla\cdot\textbf{u}\right)1\right)  to {math} \tau = 2\eta\varepsilon\left(\textbf{u}\right)  :::